Page 107 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
following integrals for a linear one-dimensional element:

2 dN i dN j 3 99
N i dl; N dl; dl; N dl; N i N j dl (3.297)
i
i
l l l dx dx l l
Exercise 3.7.4 Derive the shape functions for a one-dimensional linear element in which
both the temperature and the heat fluxes should continuously be varying in the element.
(Note that degrees of freedom for a one-dimensional linear element are T i , q i , T j , q j .)
Exercise 3.7.5 The solution for temperature distribution in a linear triangle gives the nodal
◦
◦
◦
temperature as T i = 200 C, T j = 180 C and T k = 160 C. The coordinates of i, j and k are
(x i = 2cm, y i = 2cm), (x j = 6cm, y j = 4 cm) and (x k = 4cm, y k = 6 cm). Calculate the
temperature at a location given by x = 3 cm and y = 4 cm. Calculate the coordinates of
◦
the isotherm corresponding to 170 C. Calculate the heat flux in the x and y directions if
◦
the thermal conductivity is 0.5 W/m C. Also, show that the sum of the shape functions at
(x = 3cm, y = 4 cm) is unity.
Exercise 3.7.6 For a one-dimensional quadratic element evaluate the integrals (Note : con-
vert N i , N j and N k to local coordinates and then integrate.)

N i dl; N j dl; N k dl; N i N j dl (3.298)
l l l l
Exercise 3.7.7 The nodal values for a rectangular element are given as follows,
◦
◦
x i = 0.25 cm, y i = 0.20 cm, x j = 0.30 cm, y m = 0.25 cm, T i = 150 C, T j = 120 C,
T k = 100 C, T m = 110 C Calculate (a) The temperature at the point C(x = 0.27 cm,
◦
◦
y = 0.22 cm). (b) x, y coordinates of the isotherm 130 C (c) Evaluate ∂T /∂x and ∂T /∂y
◦
at the point C.
Exercise 3.7.8 Calculate the shape functions for the six-node rectangle shown in Figure 3.31.
Exercise 3.7.9 Evaluate the partial derivatives of the shape functions at ψ = 1/4 and η =
1/2 of a quadrilateral element shown in Figure 3.32 assuming that the temperature is approx-
imated by (a) Bilinear (b) Quadratic interpolating polynomials.
Exercise 3.7.10 Calculate the derivatives ∂N 6 /∂x and ∂N 6 /∂y at the point (2, 5) for a
quadratic triangle element, when the geometry is represented by a three-node triangle as
shown in Figure 3.33.

4 5 6
3 cm

1 2 3

3 cm 3 cm
Figure 3.31 Rectangular element